Abstract. We examine the autonomous reaction-diffusion system ut = λ1uxx + f(u, v)u − v, vt = λ2vxx + f(u, v)v + u for t > 0 with Dirichlet boundary conditions on I = (0, 1), where λ1, λ2 are positive constants. If λ1 and λ2 coincide, then the zero solution is the only stationary solution. If the zero solution is unstable and f satisfies some monotonicity condition, then periodic motion can occ...

In the special case of where the spectrum σ = {λ1, λ2, λ3, 0, 0, . . . , 0} has at most three nonzero eigenvalues λ1, λ2, λ3 with λ1 ≥ 0 ≥ λ2 ≥ λ3, and λ1 + λ2 + λ3 = 0, the inverse eigenvalue problem for symmetric stochastic n × n matrices is solved. Constructions are provided for the appropriate matrices where they are readily available. It is shown that when n is odd it is not possible to re...

We consider spectral optimization problems of the form min { λ1(Ω;D) : Ω ⊂ D, |Ω| = 1 } , where D is a given subset of the Euclidean space R. Here λ1(Ω;D) is the first eigenvalue of the Laplace operator −∆ with Dirichlet conditions on ∂Ω ∩D and Neumann or Robin conditions on ∂Ω ∩ ∂D. The equivalent variational formulation λ1(Ω;D) = min {∫

We give a lower bound for the first gap λ2 − λ1 of the Dirichlet eigenvalues of the Schrödinger operator on a bounded convex domain Ω in a class of Riemannian manifolds, namely λ2 − λ1 ≥ π diameter(Ω) + ( 12 π − 1)α. For the Laplacian on disks in Rn, we have λ2 − λ1 ≥ 4 3 π diameter(Ω) .

We prove that, for any complex vector bundle E of rank e on a compact Kähler manifold X, we have that μ(SE) = |λ| μ(E) for any λ = (λ1, ..., λe−1) with λi ∈ N and λ1 ≥ ... ≥ λe−1, where |λ| = λ1 + ... + λe−1, the symbol S denotes the Schur functor and μ is the slope. This result has already been stated, without proof, by Ottaviani in 1995. AMS Subject Classification: 19L10, 55R10

We discuss some enumerational results relating the numbers of F (n;λ1, ..., λm) and F (n;λ ′ 1, ..., λ ′ k) frequency squares of order n. In particular, for any frequency vector (λ1, ..., λm) of n, we discuss some enumerational results relating the number of F (n;λ1, ..., λm) frequency squares and the number of latin squares of order n. In Section 4 we also discuss some enumerational results fo...

In this paper, we find an explicit combinatorial criterion for the existence of a nonzero GLn−1(K)-high weight vector of weight (λ1, . . ., λi−1, λi−d, λi+1, . . . , λn−1), where d < char(K) and K is an algebraically closed filed, in the irreducible rational GLn(K)-module Ln(λ1, . . ., λn) with highest weight (λ1, . . . , λn). For this purpose, new modular lowering operators are introduced.

We apply topological methods to study the smallest non-zero number λ1 in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set {S ∈ M2 : λ1(S) > 1 4 } is unbounded and disconnects the moduli spaceM2. Using this, for genus g ≥ 3, we show the existence of eigenbranches that start as λ1 and eventually becomes > 1 4 .

A group divisible design GDD(m,n; 3, λ1, λ2) is an ordered triple (V,G,B), where V is a m+ n-set of symbols, G is a partition of V into 2 sets of sizes m,n, each set being called group, and B is a collection of 3-subsets (called blocks) of V , such that each pair of symbols from the same group occurs in exactly λ1 blocks; and each pair of symbols from different groups occurs in exactly λ2 block...

Let G be a simple graph with n vertices, and let G be its complement. Let δ(G) = δ and ∆(G) = ∆ be the minimum degree and the maximum degree of vertices of G, respectively. In this paper, some upper bounds for the Laplacian spectral radius of the Nordhaus-Gaddum type are obtained as follows: λ1(G) + λ1(G) ≤ 3n+∆− δ − 5 + √ 2(n+∆)2 + 2(δ + 1)2 − 8nδ 2 , λ1(G) + λ1(G) ≤ n+∆− δ − 1 + √( 2− 1 ω(G) ...